Young Children's Understanding of the Successor Function

Young Children’s Understanding of the Successor Function

Jennifer A. Kaminski ([email protected]) Department of Mathematics and Statistics, Wright State University Dayton, OH 45435 USA

Abstract for numbers larger than one. Then children become “two- knowers”, correctly responding to requests for one or two This study examined 4-year-old children’s understanding of the successor function, the concept that for every positive items, but incorrectly for larger numbers. Subsequently, integer there is a unique next integer. Children were tested in they become “three-knowers” and possibly later “four- the context of cardinal numbers and ordinal numbers. The knower”. After several months as a “subset-knower” (for results suggest that knowledge of the successor for cardinal collections less than five), a child typically reaches a stage numbers precedes that for ordinal numbers. In addition, for where they can respond correctly to a request for any both cardinal and ordinal numbers, children generally failed number of items (i.e. cardinal-principle knower). to demonstrate understanding that the successor of a given number is unique. Some researchers have suggested that children transition from subset knowers to cardinal-principle knowers through Keywords: Counting; Natural Numbers; Cardinal Number; a bootstrapping process (Carey, 2004). Children who have Ordinal Number; Successor Function. knowledge of cardinality for collections of one, two, and three items extend this knowledge to larger collections of Introduction items by the inductive inference “If the number N represents A large amount of research has examined children’s the cardinality of a set of N items, then the next number in understanding of natural numbers (i.e. positive integers) and the count list, N+1, represents the cardinality of a set of N counting (e.g. Baroody, 1987; Fuson, 1988; Gelman & +1 items. Sarnecka and Carey (2008) describe this notion Gallistel, 1978; Piaget, 1952; Schaeffer, Eggleston, & Scott, of a “next number” representing cardinality of a set one unit 1974; Wynn, 1990, 1992). One interesting phenomenon that larger than currently under consideration as knowledge of has been observed is that children aged two to three years the mathematical concept of successor. They suggest that may appear at first glance to know how to count; they may cardinal-principle knowers have implicit knowledge of the correctly recite the count list (“one, two, three, four, …”) successor function that enables them to make the induction and even point to items in a collection and tag them with a from knowledge of cardinalities of small sets (e.g. less than number (e.g. Briars & Siegler, 1984; Fuson, 1988). four) to cardinalities of larger sets. They support their However, when asked “how many” items are in a collection argument by demonstrating that cardinal-principle knowers, they just “counted”, these children often do not know and not subset-knowers, recognize (1) that adding one item (Schaeffer, Eggleston, & Scott, 1974; Wynn, 1990, 1992). and not subtracting one item to a set of cardinality five Moreover, many children who can correctly answer the produces a set of cardinality six and (2) that adding one item “how many” question are unable to produce a collection of to a set of cardinality four results in five while adding two objects of a specific cardinality (Le Corre & Carey, 2007; items results in six. Wynn, 1992). For example, an experimenter might ask a While it is clear from these results that children who child to “give three toys to a puppet” and the child would understand the cardinality principle, as evidence by accurate give an incorrect number. Failure on these basic tasks performance on “Give-N” tasks, perform better on other suggests that young children (typically those under four numerical tasks than children who don’t understand the years of age) do not understand fundamental aspects of cardinality principle, these results do not necessarily imply natural numbers, including the Cardinality Principle. (see that cardinal-principle knowers have knowledge of the Gelman and Gallistel, 1978 for principles of counting) This successor function. The interpretation of successor that principle states that when counting a collection of items, the Sarnecka and Carey (2008) employ is essentially a notion of number associated with the final item counted represents the “there is a next”. This interpretation is inconsistent with a cardinality of the collection of items (i.e. the number of formal mathematical definition of successor function, and items). more importantly it is insufficient to generate the natural Accurate performance on such “Give-N” tasks has been numbers and hence explain the acquisition of the natural taken as evidence that children have knowledge of the numbers. As Lance, Asmuth, and Bloomfield (2005) point cardinality principle (Le Corre & Carey, 2007; Wynn, out, the simple notion that there is a “next” number can 1992). This knowledge emerges in a series of stages. First describe sets other than the natural numbers. For example, children respond by giving an arbitrary or an idiosyncratic the set of integers with addition modulo 12 (i.e. equivalent number of items (e.g. always a handful or always one). By to telling time on a standard clock face) has a successor age 2 ½ to 3 years, children typically respond correctly function; the successor of one is two, the successor of two is when asked to give one item, but give an incorrect number

1045 three, etc. However, unlike the natural numbers, the Mathematically, uniqueness can be proven by successor of twelve is one. demonstrating that for two natural numbers, a and b, if the The problem with the Sarnecka and Carey interpretation successor of a = the successor of b, then a = b. Knowledge is that they omitted two necessary characteristics of the of a unique successor can be tested in an analogous way by successor function. First, that the number one is not the stating a number and asking the participant what number successor of any number (if we do not include zero in the immediately precedes it. Two testing contexts were created, set of natural numbers). Second, the successor of a number one cardinal and one ordinal. Participants were tested on is unique. For example, the successor of two is only three; their ability to state a next number when given a number and three is the successor only of two. Specifically, the (i.e. successor). These questions are denoted as +1 natural numbers can be defined by the Dedekind/ Peano’s questions in the Method section. Participants were also Axioms that state the following (Dedekind, 1901). tested on their ability to state a preceding number when 1. One is a natural number. given a number (i.e. uniqueness of the successor). These 2. Every natural number has a successor that is a natural questions are denoted as -1 questions in the Method section. number. The cardinal context involved collections of circular disks 3. One is not the successor of any natural number. that were placed under a cup. Participants needed to state 4. Two natural numbers for which the successors are how many disks were under the cup when a disk was added equal are themselves equal (i.e. the successor of any or removed. The ordinal context involved a simple board natural number is unique). game in which participants were asked the number 5. If a set, S, of natural numbers contains one and for associated with locations before or after the location of a every k in S, the successor of k is also in S, then every game pawn (see Figure 1). natural number is in S. These axioms provide necessary and sufficient criteria to Experiment generate the set of natural numbers and therefore knowledge of these axioms could explain the acquisition of natural Method numbers. Participants Participants were 35 preschool children (21 The interpretation of successor used in the Sarnecka and girls, 14 boys, M = 3.86 years, SD = 0.15) recruited from Carey (2008) study does not demonstrate uniqueness of the preschools and childcare centers in the Columbus, Ohio area. successor function. Specifically, demonstrating that a child Materials and Design Participants were given a series of recognizes five as the cardinality of a set that contained four different tasks, How-Many task, Give-N task, Cup task, and and then another item was added, does not imply that the Game task. child knows that cardinality of five cannot be achieved by How-Many task. This task consisted of six questions and adding one to another set size. In our everyday lives, as involved black circular chips approximately 1 inch in adults we take this implication for granted, but we cannot diameter. For each question, the experimenter placed a assume that children do. Therefore demonstrating that collection of chips on a plastic plate approximately 7 inches children have knowledge of the successor without in diameter. The experimenter then showed the child the demonstrating its uniqueness cannot fully explain how collection of chips and asked, “Can you tell me how many children acquire knowledge of natural numbers. chips are here?” The questions presented collections of size Additionally, the concept of successor is an ordinal 2, 3, 4, 5, 6, and 8 in a pseudo-random order. concept, yet is has been tested in the context of cardinality. Give-N task. This task consisted of six questions testing the The goal of the present study was to examine the numbers 2, 3, 4, 5, 6, and 8 in a pseudo-random order. The conditions under which young children demonstrate experimenter placed a collection of ten chips in front of the knowledge of the successor function. If children who child. The experimenter then gave the child the plastic plate understand the cardinality principle demonstrate knowledge and asked, “Can you give me N chips on this plate?” of a unique successor for cardinal numbers, then perhaps, as Cup task (Cardinal Test). This task was designed to Sarnecka and Carey suggest, knowledge of the successor measure knowledge of the successor function in the context function “turns a subset knower into a cardinal principle of cardinal numbers. Materials for this task included black knower”. Furthermore, if children demonstrate this circular chips and a red, plastic cup. For each question, the knowledge in the context of both cardinal number and experimenter placed a specific number of chips in front of ordinal numbers, then knowledge of the successor may the child and clearly stated the number of chips presented. explain the acquisition of natural number and their ordinal Next, the experimenter covered the chips with the cup and properties. did one of two things. Either the experimenter added However, if children fail to appreciate the uniqueness of another chip to the hidden amount or took a chip away from the successor, then we cannot conclude that children’s the hidden amount. To prevent the child from seeing the understanding of the successor is sufficient to give rise to adjusted number of chips, the chip was added or removed by knowledge of the cardinality principle and to the natural lifting a corner of the cup only slightly and sliding a chip in numbers. or out. After the adjustment was made, the child’s job was to tell the experimenter the number of chips under the cup.

1046 Two practice trials with feedback were given before the test questions. A total of twenty-two questions were presented. There were ten questions in which one chip was added (i.e. Cardinal +1 questions), two for each initial number of 1 through 5. For example, for an initial number of 1, one chip was placed under the cup and then an additional chip was added. There were eight questions in which one chip was removed (i.e. Cardinal -1 questions), two for each initial number of 2 through 5. For example, for an initial number Figure 1. Picture of the game board and pawn used to of 2, two chips were placed under the cup and then one chip test knowledge of the successor function in the context removed. There were four additional questions in which of ordinal numbers. two chips were added or removed (i.e. 2+2, 3+2, 3-2, and 4- 2). Questions were presented in a pseudo-random order. leaving her pawn on the stopping location. The following Board Game task (Ordinal Test). This task was designed to is an excerpt of the script. measure knowledge of the successor function in the context I am counting in my head. You don’t know where I of ordinal numbers. Prior to administering the actual started, but I am stopping right here (placing the game successor game task, participants were given a short test of pawn on a specific space on the board) at this number, X understanding of the board game. This test was administered (where X ranged from 1 to 6). What number did I stop to ensure that participants understood how to move a game at? pawn across spaces on a board by counting contiguous Corrective feedback was given if necessary. Then the spaces. experimenter asked one of two questions, either “What Materials for this task consisted of a laminated “game number would I count if I kept counting until here board” with ten contiguous, linear spaces and several colored (pointing to the next space forward)” or “What number did game pawns (see Figure 1). Three arrows were placed on the I count when I was here (pointing to the space before the board game to guide the direction of movement along the stopping space)?” Twenty-two questions were presented in spaces on the game board. For half the participants the a pseudo-random order. These questions were analogous to arrows pointed left, and for the other half the arrows pointed those in the Cardinal Test (i.e. cup task) and included right. The experimenter told the child that they were going to Ordinal +1 questions, Ordinal -1 questions, and four play a number game. Both the child and the experimenter additional questions in which two chips were added or chose a game pawn to use. The experimenter told the child removed (i.e. 2+2, 3+2, 3-2, and 4-2). that he/she can count on the board by counting and moving Prior to the actual ordinal test just described, the the pawn one space at a time in the direction of the arrows on experimenter gave the participant four practice questions the board. The experimenter demonstrated this and then told with corrective feedback. For two of the questions, the the child, “the funny thing about this game is that you never experimenter actually indicated her starting position and know where I am going to start my counting”. The counted out loud. For two questions, the experimenter did experimenter showed the child two examples. The following not indicate her starting position and did not count out is an excerpt of the script. loud. I might start counting on any one of these spaces. So for Participants were randomly assigned to one of two example, maybe I would start here (placing the game between-subjects condition that differed in the order of two piece on the 3rd space on the board). If I count to 5, I successor tasks (i.e. the cup task and the game task). would count like this; count 1, 2, 3, 4, 5 (tapping on each subsequent space with the game piece) and stop here, at Procedure Participants were tested in a quiet room at their 5. preschool by a female experimenter. The experiment was After the two examples, the participant was given five conducted in two sessions over two days. In the first questions to measure understanding of the game (i.e. session, participants completed the How-many task, the practice game test). For these questions, the experimenter Give-me task, and one of the successor tasks. In the second pointed to a specific space on the board, asked the child to session, participants completed the second successor task. count from that space to a given number (2 through 6 in Responses were recorded on paper by the experimenter. pseudo-random order), and leave his/her pawn on the stopping space. The child was told to use his/her game Results pawn and tap each passing space as he/ she counted. Each Each participant was categorized as a cardinal principle of these trials had a different starting space. knower (CP-knower) or non-cardinal principle knower For the ordinal successor test, the experimenter (Non-CP-knower) based on his/her performance on the explained to the participant that in the game the child will “How Many” and “Give-N” questions. A participant was not know the space on which the experimenter will start considered to be a CP-knower if he/she achieved scores of counting, the experimenter will count in her head, not out at least 83% correct (i.e. 5/6) on both the “How many” and loud, and then tell the child the number that she stopped at, the “Give-N” tests. The criteria for knowledge of the

1047 cardinality principle were similar to that of previous research (e.g. Sarnecka & Carey, 2008; Wynn, 1992). Seventeen participants were categorized as CP-knowers and 18 participants were categorized as Non-CP-Knowers. Unlike previous research, the present analysis did not examine performance of specific subset-knowers. The present analysis focused on performance on the successor tasks (i.e. Cardinal +1, Cardinal -1, Ordinal +1, and Ordinal -1 questions). Tasks similar to the cardinal successor task have been used in previous research (e.g. Sarnecka & Carey, 2008). However, the ordinal successor task has not been used in previous research. Therefore, to ensure that any poor Figure 2. Mean Accuracy (% Correct) on Cardinal and performance on this task was due to lack of knowledge of Ordinal +1 Questions. Error bars represent standard error the successor function and not due to failure to understand of the mean. the task, the primary analysis included only those participants who demonstrated understanding of the context of the game (the ordinal successor task) by scoring at least 4 of 5 correct on the practice game test. Seven children were excluded from the analysis for scoring below 4 (M = 1.00, SD = 1.15). Accuracy on the practice game test was high for the participants who scored above 4 (M = 4.89, SD = .31). There were differences between the seven excluded participants and the remaining participants in the frequency of CP-knowers. Only one of the excluded seven participants met the criteria for CP-knower, while 16 of 28 (57%) of the remaining participants were CP-knowers, this difference in proportion was significant, Fisher Exact Test, p = .05. Also note that for these participants, the mean scores on the Figure 3. Mean Accuracy (% Correct) on Cardinal and Cardinal +1 questions (M = 58.6%, SD = 20.7%) and on the Ordinal -1 Questions. Error bars represent standard error Cardinal -1 questions (M = 64.3%, SD = 26.4%) was not of the mean. statistically different from those of the participants who did understand the game (M = 72.9%, SD = 19.8% and M = Cardinal -1 question and each Ordinal -1 question for Non- 71.9%, SD = 18.2% for the Cardinal +1 and Cardinal –1 CP-knowers and CP-knowers. Both CP-knowers and Non- questions respectively), independent samples t(33)s < 1.74, CP-knowers were more accurate on cardinal questions than ps > .09. In addition, there was no difference in age ordinal questions. Scores on Cardinal +1 questions were between these groups of children (M = 3.8 years SD = .17 higher than scores on Ordinal +1 questions, paired sample ts and M = 3.9 years SD = .14 for the seven excluded > 4.63, ps < .001; scores on Cardinal -1 questions were participants and the remaining participants respectively). higher than scores on Ordinal -1 questions, paired sample ts The remaining analysis focuses on performance on the > 5.63, ps < .001. However, beyond this comparison, four types of successor questions (i.e. Cardinal +1, Cardinal different patterns of results emerged for CP-knowers and -1, Ordinal +1, and Ordinal -1). No differences in Non-CP-knowers. performance between the two between-subject conditions On both Cardinal +1 questions and Ordinal +1 questions st st (i.e. Cup 1 and Game 1 ) was found for any of these (Figure 2), CP-knowers were more accurate than Non-CP- measures, independent samples t(26)s < 1.15, ps > .25. knowers, independent samples t(26)s > 2.06, ps < .05. On There was also no difference between the two conditions in Cardinal +1 questions, Non-CP-knowers had a clear drop in st proportion of CP-knowers (60% of Cup 1 participants and accuracy as the test numbers increased. Repeated Measures 54% of Game 1st participants), �! 1, � = 28 = .11, � = ANOVA shows significant differences in accuracy on the .74. There was no difference in performance on the Ordinal different test questions, F(4, 44) = 4.99, p < .01, �! = .31 +1 and Ordinal -1 tests for participants who moved the and downward trend in accuracy, linear and cubic contrasts game pawn to the left and those who moved the game pawn F(1, 11)s > 6.40, ps < .03, �!s = .36. CP-knowers did not to the right, independent samples t(26)s < .13, ps > .89. have a significant drop in performance on the Cardinal +1 Therefore, data was collapsed across these conditions. question, repeated measures ANOVA, F(4, 60) = .98 p > Figure 2 presents mean accuracy on each Cardinal +1 .42. On the Ordinal +1 questions for Non-CP-knowers, question and each Ordinal +1 question for Non-CP-knowers there was a moderate difference in accuracy across the and CP-knowers. Figure 3 presents mean accuracy on each different test numbers, repeated measures ANOVA F(4, 44) = 2,37 p =.06, �! = .18. No significant difference in

1048 accuracy on the Ordinal +1 questions was found across the Table 1: Mean Accuracy (% Correct) for +2 and -2 different test numbers for CP-knowers, repeated measures Questions. Standard deviations are in parentheses. ANOVA F(4, 60) = 1.97 p =.11. While CP-knowers were more accurate than Non-CP- Non CP- CP- knowers on the Cardinal +1 and Ordinal +1 questions, there knower knower were no differences in accuracy between these two groups Cardinal Questions on Cardinal -1 and Ordinal -1 questions (Figure 3), + 2 42 (36) 56 (40) independent samples t(26)s < 1.34, ps > .19. Both CP- knowers and Non-CP-knowers had a linear drop in accuracy - 2 79 (26) 78 (26) on Cardinal -1 questions as the test number increased, Ordinal Questions repeated measures ANOVA Fs > 6.27 ps < .001, �!s > .29, + 2 8 (19) 22 (36) ! linear contrasts Fs > 17.9 ps < .01, � s > .59. On Ordinal - - 2 17 (25) 13 (29) 1 questions, there were no differences in accuracy across questions for either the CP-knowers or Non- CP-knowers, repeated measures ANOVAs Fs < .26 ps > .85. The fact that accuracy on Cardinal -1 questions drops There were also no significant differences in accuracy linearly as the test number increases suggests that an between CP-knowers and Non-CP-knowers on the Cardinal awareness of unique successors may emerge in stages with and Ordinal +2 and -2 questions (see Table 1), independent smaller numbers preceding larger numbers. Therefore samples ts <1.16, ps > .25. Both CP-knowers and Non-CP- young children’s understanding of unique successors is not knowers were more accurate on Cardinal questions than on an understanding of a general principle, but rather it is Ordinal questions, paired samples ts > 2.34, ps < .04. isolated and tied to specific numbers. Additionally, Non-CP-knowers scored higher on Cardinal It is also interesting to note that overall, CP-knowers +2 questions than on Cardinal -2 questions, paired sample outperformed Non-CP-knowers only on the Cardinal and t(11) = 3.00, p < .02. CP-knowers were moderately more Ordinal +1 questions. This suggests that knowledge of the accurate on Cardinal +2 questions than on Cardinal -2 cardinality principle provides no better insight into the questions, t(15) = 1.81, p = .09. Scores on the Ordinal +2 uniqueness of the successor (Cardinal and Ordinal -1 and Ordinal -2 were quite low with no differences between questions) and no better accuracy on the +2 and -2 questions +2 and -2 questions for either group, paired samples ts < than the absence of this knowledge. 1.15, ps > .27. The present findings also suggest that acquisition of cardinality precedes that of ordinality (see Colome & Noel, Discussion 2012 for similar findings). The fact that young children can The goal of the present study was to investigate young correctly recite the count list might suggest that they have children’s understanding of natural numbers by examining an appreciation of order and an understanding of successor the conditions under which they demonstrate knowledge of in the context of order. However, in the present study, the successor function. Specifically, do children have participants could not reliably determine successors in a knowledge of the successor function that can explain the simple ordinal number task. This indicates that the count emergence of the cardinality principle? Do children have an list that children recite is simply a memorized sequence that understanding of the successor function in both cardinal and does not reflect the properties of natural numbers. At the ordinal contexts? same time, both CP-knowers and Non-CP-knowers were The results indicate that CP-knowers have some more able to determine the successor in the cardinal number understanding of the concept of a successor that Non-CP- task than in the ordinal task. Performance on the +2 and -2 knowers do not have. When asked to state a successor of a questions provides additional evidence that knowledge of given number (i.e. +1 questions), CP-knowers outperformed cardinal numbers precedes that of ordinal numbers; both Non-CP-knowers in both cardinal and ordinal contexts. CP- CP-knowers and Non-CP-knowers were markedly more knowers were quite accurate stating a successor for cardinal accurate on the cardinal questions than the ordinal numbers (M = 80.0, SD = 7.5 on all Cardinal +1 questions), questions. but their accuracy was much lower for ordinal numbers (M The results of this study demonstrate that young children = 51.9, SD = 8.7 on all Ordinal +1 questions). do have some knowledge of the successor, but the nature of However, the results provide no evidence that CP- this knowledge is not sufficiently constrained to explain the knowers appreciate uniqueness of the successor function. emergence of the cardinality principle and full They could not reliably state a correct preceding number (-1 understanding of the natural numbers. What knowledge or questions). In both the cardinal and the ordinal contexts, experience does provide sufficient constraints to account for their accuracy on stating a preceding number was no higher mature representations of natural numbers is unclear and than that of the Non-CP-knowers. Therefore, the extent of requires further investigation. CP-knowers’ knowledge of the successor cannot explain the acquisition of the cardinality principle and of natural numbers.

1049 Acknowledgments This research was supported by a grant from the Institute of Education Sciences, U.S. Department of Education (#R305B070407).

References Baroody, A. J. (1987). Children’s mathematical thinking: A developmental framework for preschool, primary, and special education teachers. New York: Columbia University Teachers College. Briars, D., & Siegler, R. S. (1984). A featural analysis of preschoolers’ counting knowledge. Developmental Psychology, 20, 607–618. Carey, S. (2004). Bootstrapping and the origins of concepts. Daedalus, 133, 59–68. Colome, A., & Noel, M. P. (2012). One first? Acquisition of the cardinal and ordinal uses of numbers in preschoolers. Journal of Experimental Child Psychology, 113, 233-247. Dedekind, R. (1901). The nature and meaning of numbers (W. W. Beman, trans.). In Essays on the theory of numbers. London: Open Court. Available online: https://openlibrary.org/works/OL2409707W/Essays_in_th e_theory_of_numbers_1._Continuity_of_irrational_numb ers_2 Fuson, K. C. (1988). Children’s counting and concepts of number. New York: Springer-Verlag. Gelman, R., & Gallistel, C. R. (1978). The child’s understanding of number. Cambridge, MA: Harvard University Press. Le Corre, M., Van de Walle, G., Brannon E. M., & Carey, S. (2006). Re-visiting the competence/performance debate in the acquisition of the counting principles. Cognitive Psychology, 52, 130-169. Piaget, J. (1952). The child’s conception of number. Routledge & Kegan Paul: London, England. Rips, L. J., Asmuth, J., & Bloomfield, A. (2006). Giving the boot to the bootstrap: How not to learn the natural numbers. Cognition, 101, B51-B60. Sarneka, B.W., & Carey, S. (2008). How counting represents number: What children must learn and when they learn it. Cognition, 108, 662-674. Schaeffer, B., Eggleston, V. H., & Scott, J. L.(1974). Number development in young children. Cognitive Psychology, 6, 357-379. Wynn, K. (1990). Children’s understanding of counting. Cognition, 36, 155–193. Wynn, K. (1992). Children’s acquisition of number words and the counting system. Cognitive Psychology, 24, 220– 251.

1050